# Model Academic Standards for Mathematics

## Glossary of Terms

**Algorithm. An** established step-by-step procedure
used to achieve a

desired result. For example, the addition algorithm for the sum of two

two-digit numbers where carrying is required:

**Arbitrary unit (of measure).** A unit that is not
part of the standardized metric or US

Customary systems. For example, using one’s own shoe size to measure the length
of a door

opening or saying that the area of an exhibition hall floor is “about the size
of two football

fields.”

**Associative property.** When adding or multiplying
three numbers, it doesn’t matter if the

first two or the last two numbers are added or multiplied first . For example,

**Attribute (measurable)**. An identifiable property of
an object, set, or event that is subject

to being measured. For example, some of the measurable attributes of a box are
its length,

weight, and capacity (how much it holds).

**Box plot.** A graphic method that shows the
distribution of a set of data by using the

median, quartiles, and the extremes of the data set. The box shows the middle 50
percent of

the data; the longer the box, the greater the spread of the data.

**Central tendencies. **A number which in some way
conveys the “center” or “middle” of a

set of data. The most frequently used measures are the mean and the median.

**Combinations. **Subsets chosen from a larger set of
objects in which the order of the items

in the subset does not matter. For example, determining how many different
committees of

four persons could be chosen from a set of nine persons. (See also,
Permutations)

**Commutative property.** Numbers can be added or
multiplied in either order. For

example, 15 + 9 = 9 + 15; 3 x 8 = 8 x 3.

**Congruence.** The relationship between two objects
that have exactly the same size and

shape.

**Correlation. **The amount of positive or negative
relationship existing between two

measures. For example, if the height and weight of a set of individuals were
measured, it

could be said that there is a positive correlation between height and weight if
the data

showed that larger weights tended to be paired with larger heights and smaller
weights

tended to be paired with smaller heights. The stronger those tendencies, the
larger the

measure of correlation.

**Deciles.** The 10th, 20th, 30th, ...90th percentile
points. (See definition for Percentile)

**Direct measurement.** A process of obtaining the
measurement of some entity by reading a

measuring tool, such as a ruler for length, a scale for weight, or a protractor
for angle size.

**Dispersion.** The scattering of the values of a
frequency distribution (of data) from an

average.

**Distributive property. **Property indicating a
special way in which multiplication is

applied to addition of two (or more) numbers. For example,

5 x 23 = 5 x (20 + 3) = 5 x 20 + 5 x 3 = 100 + 15 = 115.

**Expanded notation.** Showing place value by
multiplying each digit in a number by the

appropriate power of 10. For example, 523 = 5 x 100 + 2 x 10 + 3 x 1 or 5 x 10^{2}
+ 2 x 10^{1} + 3

x 10^{0}.

** Exponential function .** A function that can be represented by an equation of the
form

y = ab^{x} + c, where a, b, and c are arbitrary, but fixed, numbers and a ≠ 0 and b
> 0 and b ≠ 1.

**Exponential notation (exponent).** A symbolic way of showing how many times a
number

or variable is used as a factor. In the notation 53, the exponent 3 shows that 5
is a factor

used three times ; that is 5^{3} = 5 x 5 x 5 = 125.

**Frequency distribution.** An organized display of a set of data that shows how
often each

different piece of data occurs.

**Function.** A relationship between two sets of numbers or other mathematical
objects

where each member of the first set is paired with only one member of the second
set.

Functions can be used to understand how one quantity varies in relation to (is a
function of)

changes in the second quantity. For example, there is a functional relationship
between the

price per pound of a particular type of meat and the total amount paid for ten
pounds of

that type of meat.

**Functional notation. **A convenient way to show how a function takes a value and

transforms it into a new value. A commonly used notation is called Euler
notation and is

written f (x) and read “f of x.” For example, if f (x) = 3x + 5, this means that
the function f

will take any value from the domain of x and multiply it by 3 and then add 5 to
make the

new value for x (e.g., f (4) = 3*4 + 5 = 17).

**Identity.** For addition: The number 0; that is N + 0 = N for any number N. For

multiplication: The number 1; that is, N x 1 = N for any number N.

**Indirect measurement. **A process where the measurement of some entity is not
obtained

by the direct reading of a measuring tool or by counting of units superimposed
alongside or

on that entity. For example, if the length and width of a rectangle are
multiplied to find the

area of that rectangle, then the area is an indirect measurement.

**Integers. **The set of numbers: {..., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,
6,...}

**Intercept. **The points where a line drawn on a rectangular-coordinate-system
graph

intersect the vertical and horizontal axes.

**Inverse.** For addition: For any number N, its inverse (also called opposite) is a
number -N

so that N + (-N) = 0 (e.g., the opposite of 5 is -5, the opposite of -3/4 is
3/4).

For multiplication: For any number N, its inverse (also called reciprocal) is a
number N* so

that N x (N*) = 1 (e.g., the reciprocal of 5 is 1/5; the reciprocal of -3/4 is
-4/3).

**Line of best fit.** A straight line used as a best

approximation of a summary of all the points in a

scatter-plot* (See definition below). The position and

slope of the line are determined by the amount of

correlation* (See definition above) between the two

paired variables involved in generating the scatter-plot.

This line can be used to make predictions about the

value of one of the paired variables if only the other

value in the pair is known

**Line plot.** A graphical display of a set of data where

each separate piece of data is shown as a dot or mark

above a number line.

** Linear equation .** An equation of the form y = ax + b, where a and b can be any
real

number. When the ordered pairs (x, y) that make the equation true for specific
assigned

values of a and b are graphed, the result is a straight line.

**Matrix (pl.: matrices).** A rectangular array of numbers, letters, or other
entities arranged

in rows and columns.

**Maximum/minimum (of a graph).** The highest/lowest point on a graph. A relative

maximum/minimum is higher/lower than any other point in its immediate vicinity.

**Mean.** The arithmetic average of a set of numerical data.

**Median.** The middle value of an ordered set of numerical data. For example, the
median

value of the set {5, 8, 9, 10, 11, 11,13} is 10.

**Mode.** The most frequently occurring value in a set of data. For example, the
mode of the

set {13, 5, 9, 11, 11, 8, 10} is 11.

**Model (mathematical).** A [verb] and a noun. [Generate] a mathematical
representation

(e.g., number, graph, matrix, equation(s), geometric figure) for real-world or
mathematical

objects, properties, actions, or relationships.

**(Non)-Linear functional relationship.** (See definition of Function above.) Many

functions can be represented by pairs of numbers. When the graph of those pairs
results in

points lying on a straight line, a function is said to be linear. When not on a
line, the

function is nonlinear.

**Nonroutine problem.** A problem is nonroutine if it cannot be solved simply by

substituting specific data into a formula that the student knows, or by
following step-bystep

the solution of a type of problem with which the student is already familiar.

**Outlier.** For a set of numerical data, any value that is markedly smaller or
larger than

other values. For example, in the data set {3, 5, 4, 4, 6, 2, 25, 5, 6, 2} the
value of 25 is an

outlier.

**Patterns.** Recognizable regularities in situations such as in nature, shapes,
events, sets of

numbers. For example, spirals on a pineapple, snowflakes, geometric designs on
quilts or

wallpaper, the number sequence {0, 4, 8, 12, 16,...}.

**Percentile.** A value on a scale that indicates the percent of a distribution that
is equal to it

or below it. For example, a score at the 95th percentile is equal to or better
than 95 percent

of the scores.

**Permutations.** Possible arrangements of a set of objects in which the order of
the

arrangement makes a difference . For example, determining all the different ways
five books

can be arranged in order on a shelf.

**Prime number.** A whole number greater than 1 that can be divided exactly (i.e.,
with no

remainder) only by itself and 1. The first few primes are 2, 3, 5, 7, 11, 13,
17, 19, 23, 29, 31,

37….

**Pythagorean theorem (relationship).** In a right triangle, c^{2} = a^{2} + b^{2}, where c
represents

the length of the hypotenuse (the longest side of the triangle which is opposite
the right

angle), and a and b represent the lengths of the other two, shorter sides of the
triangle.

**Quadratic function.** A function that can be represented by an equation of the
form y = ax^{2}

+ bx + c, where a, b, and c are arbitrary, but fixed, numbers and a ≠ 0. The
graph of this

function is a parabola .

**Quartiles.** The 25th, 50th and 75th percentile points. (See definition of
Percentile.)

**Range** (of a set of data). The numerical difference between the largest and
smallest values

in a set of data.

**Rational number.** A number that can be expressed as the ratio, or quotient, of
two

integers, a/b, provided b ≠ 0. Rational numbers can be expressed as common
fractions or

decimals , such as 3/5 or 0.6. Finite decimals, repeating decimals, mixed numbers
and whole

numbers are all rational numbers. Nonrepeating decimals cannot be expressed in
this way,

and are said to be irrational.

** Real numbers . **All the numbers which can be expressed as decimals.

**Real-world problems. **Quantitative and spatial problems that arise from a wide
variety of

human experiences, applications to careers. These do not have to be highly
complex ones

and can include such things as making change, figuring sale prices, or comparing
payment

plans.

**Rectangular coordinate system.** This system uses two (for a plane) or three (for
space)

mutually perpendicular lines (called coordinate axes) and their point of
intersection (called

the origin) as the frame of reference. Specific locations are described by
ordered pairs or

triples (called coordinates) that indicate distance from the origin along lines
that are

parallel to the coordinate axes.

**Scaling (Scale drawing).** The process of drawing a figure either enlarged or
reduced in

size from its original size. Usually the scale is given, as on a map 1 inch
equals 10 miles.

**Scatter plot. **Also known as scattergram or scatter diagram. A

two dimensional graph representing a set of bi-variate data.

That is, for each element being graphed, there are two separate

pieces of data. For example, the height and weight of a group of

10 teenagers would result in a scatter plot of 10 separate points

on the graph.

**Scientific notation. **A short-hand way of writing very large or very small
numbers. The

notation consists of a decimal number between 1 and 10 multiplied by an integral
power of

10. For example, 47,300 = 4.73 x 10^{4}; 0.000000021 = 2.1 x 10^{-8}.

**Similarity. **The relationship between two objects that have exactly the same
shape but not

necessarily the same size.

**Simulation.** Carrying out extensive data collection with a simple, safe,
inexpensive, easyto-

duplicate event that has essentially the same characteristics as another event
which is of

actual interest to an investigator. For example, suppose one wanted to gather
data about

the actual order of birth of boys and girls in families with five children
(e.g., BBGBG is one

possibility). Rather than wait for five children to be born to a single family,
or identifying

families that already have five children, one could simulate births by
repeatedly tossing a

coin five times. Heads vs. tails has about the same chance of happening as a boy
vs. a girl

being born.

**Slope.** A measure of the steepness or incline of a straight line drawn on a rectangularcoordinate-

system graph. The measure is obtained by the quotient “rise/run” (vertical

change divided by horizontal change) between any two points on that line.

**Stem-and-leaf plot.** A way of showing the distribution of a

set of data along a vertical axis. The plot at right shows the

data 13, 19, 33, 26, 19, 22, 34, 16, 28, 34. The ten’s digits of

these data are the stems and the one’s digits are the leaves.

1|3699

2|268

3|344

Key: 1|5 means 15

**Summary statistics.** A single number representation of the characteristics of a
set of data.

Usually given by measures of central tendency and measures of dispersion
(spread).

**Symmetry. **A figure has symmetry if it has parts that correspond with each other
in terms

of size, form, and arrangement. For example, a figure with line (or mirror)
symmetry has

two halves which match each other perfectly if the figure is folded along its
line of

symmetry.

**Transformation. **A change in the size, shape, location or orientation of a
figure.

**Transitive property.** For equality: If a = b and b = c, then a = c.

For inequality: If a > b and b > c, then a > c; or If a < b and b < c, then a <
c.

**Tree diagram. **A schematic way of showing the

number of ways a compound event may occur.

For example, the tree diagram at the right shows

the eight possible ways the tossing of three coins

could happen.

**Unit fraction. **A fraction with a numerator of 1, such as
1/4 or 1/7.

**Variable.** A quantity that may assume any one of a set of values. Usually
represented in

algebraic notation by the use of a letter. In the equation y = 2x + 7, both x
and y are

variables.

**Variance. **The value of the standard deviation squared.

**Vertical angles.** The pair of angles that are directly

across from each other when two straight lines

intersect. Angles a and b at the right are an example of

vertical angles.

**Whole numbers. **The numbers: 0, 1, 2, 3, 4, 5,….

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